Complex Analysis

10 credits

Syllabus, Bachelor's level, 1MA022

A revised version of the syllabus is available.

Code: 1MA022
Education cycle: First cycle
Main field(s) of study and in-depth level: Mathematics G2F
Grading system: Pass with distinction (5), Pass with credit (4), Pass (3), Fail (U)
Finalised by: The Faculty Board of Science and Technology, 15 March 2007
Responsible department: Department of Mathematics

Entry requirements

Several Variable Calculus

Learning outcomes

In order to pass the course (grade 3) the student should be able to

give an account of important concepts and definitions in the area of the course;

exemplify and interpret important concepts in specific cases;

formulate important results and theorems covered by the course;

describe the main features of the proofs of important theorems;

express problems from relevant areas of applications in a mathematical form suitable for further analysis;

use the theory, methods and techniques of the course to solve mathematical problems;

present mathematical arguments to others.

Higher grades, 4 or 5, require a higher level of proficiency. The student should be able to solve problems of greater complexity, i.e. problems requiring a combination of ideas and methods for their solution, and be able to give a more detailed account of the proofs of important theorems and by examples and counter-examples be able to motivate the scope of various results. Requirements concerning the student's ability to present mathematical arguments and reasoning are greater.

Content

Complex numbers, topology in C. Functions of one complex variable, limits, continuity and differentiability. The Cauchy-Riemann equations with consequences. Analytic and harmonic functions. Conformal mappings. Elementary functions from C to C, in particular Möbius transformations and the exponential function, and their mapping properties. Solution of boundary value problems in the plane for the Laplace equation using conformal mappings. Complex integration. Cauchy's integral theorem and integral formula with consequences. The maximum principle for analytic and harmonic functions. Conjugate harmonic functions. Poisson's integral formula. Uniform convergence and analyticity. Power series. Taylor and Laurent series with applications. Zeros and isolated singularities. Residue calculus with applications. The argument principle and Rouché's theorem. Briefly about connections with Fourier series and Fourier integrals.

Instruction

Lectures and problem solving sessions.

Assessment

Written and, possibly, oral examination at the end of the course. Moreover, compulsory assignments may be given during the course